10 
3* Least Squares Adjustment 
The system of observation equations 2.10 can be written more simply as: 
QV+N.A+W= 0 ... 2.11 
The least squares adjustment consists of determining from all matrices 
V and A_ , those which satisfy equation 2.11 and which also result in 
minimization of the squares of residuals, V as expressed in the form 
U = V T P V ... 2.12a 
where P is a matrix of weights for the measured plate coordinates. Then 
2 
the normal equations for the least squares adjustment of the observed 
quantities (measured plate coordinates) in equations 2.11 are: 
N^QPf 1 ^)" 1 N A_ + N T (§if 1 C£ T r 1 W =0 ... 2.12 
which may be written in a more compact form as 
J A + j = 0 ... 2.13 
in which 
a = 
m 
N (Q P 
sb' 1 
(a) 
A = 
_a_ N 
(b) 
JL = 
a W 
(c) 
so that solution to the normal equations is given by: 
A = J.' 1 j_ 
... 2.14 
... 2.15 
Plate coordinate measurement residuals, V are found by first 
calculating the Lagrange multipliers K using 
K = - (Q P _1 Q T ) _1 (N A + W) ... 2.16 
which are then substituted into equation 2.17 
V = p" 1 Q T K ... 2.17 
Formation of the normal equations and determination of plate 
coordinate residuals are simplified by taking advantage of the symmetry of 
the Q matrix in the observation equations 2.10 . First consider formation 
of the normals and solution for the corrections A^ • 
Partial normal equations, formed for condition equations required for